TFAE Type Checker: Understanding the Grammar and Data Types - Prof. Matt Flatt, Study notes of Programming Languages

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TFAE Grammar

::=^

| {+^ } | {-^ } | | {fun { :

} }

|^ { }

^

::=^ num^ |^ bool^ |^ ( -> )

TFAE Expressions and Types^ (define-type

TE

[numTE][boolTE][arrowTE

(arg

: TE)(result^ : TE)])

(define-type

Type

[numT][boolT][arrowT

(arg

: Type)(result^ : Type)])

(define-type

TypeEnv

[mtEnv][aBind

(name

: symbol)

(type

: Type)

(rest

: TypeEnv)])

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [num^ (n) ...])))

Γ^ :

num

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [num^ (n) (numT)])))

Γ^ :

num

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [add^ (l r)(type-case

Type^

(typecheck

l env)

[numT^

()...^ (typecheck

r env) ...]

[else^

(type-error

l "num")])])))

Γ^ e

:^ num 1

Γ^ e

:^ num 2

Γ^ {+

ee^1

}^ :^2

num

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [add^ (l r)(type-case

Type^

(typecheck

l env)

[numT^

()(type-case

Type^

(typecheck r env) [numT^

()^ (numT)] [else^

(type-error

r "num")])]

[else^

(type-error

l "num")])])))

Γ^ e

:^ num 1

Γ^ e

:^ num 2

Γ^ {+

ee^1

}^ :^2

num

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [id^ (name) (get-type

name^

env)])))

[ ...^

←τ^ ... ]

:

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [fun^ (name

te^ body)...])))^ Γ

[^

←τ]^1

e^ :^ τ

2

Γ^ {fun { :

τ}^ e^1

}^ :^ (

τ→^ τ^1

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [fun^ (name

te^ body)(local^ [(define

arg-type

(parse-type

te))]

...^ (typecheck

body^

(aBind

namearg-typeenv))

...)])))

Γ[^

←τ]^1

e^ :^ τ

2

Γ^ {fun { :

τ}^ e^1

}^ :^ (

τ→^ τ^1

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [fun^ (name

te^ body)(local^ [(define

arg-type

(parse-type

te))]

(arrowT

arg-type(typecheck

body^

(aBind name

arg-typeenv))))])))

Γ[^

←τ]^1

e^ :^ τ

2

Γ^ {fun { :

τ}^ e^1

}^ :^ (

τ→^ τ^1

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [app^ (fn

arg)... (typecheck

fn env) ... ...^ (typecheck

arg env) ...])))

Γ^ e

:^ (τ 1

→^ τ) 2 3

Γ^ e

:^ τ 2 2

Γ^ {

ee }^1

:^ τ^3

TFAE Type Checker

typecheck

: (FAE

TypeEnv

-> Type)

(lambda

(fae^

env) (type-case

FAE^ fae [app^ (fn

arg)(type-case

Type^

(typecheck

fn env)

[arrowT

(arg-type

result-type) ...^ (typecheck

arg^ env) ...]

[else^

(type-error

fn "function")])])))

Γ^ e

:^ (τ 1

→^ τ) 2 3

Γ^ e

:^ τ 2 2

Γ^ {

ee }^1

:^ τ^3

Pairs

{with {pair

: (num

->^ (num

->^ (num

-> num)))

{fun^ {x : num}{fun

{y : num}{fun^

{s : num}{if0^ s x y}}}}}

{with {fst

: ((num

->^ num) ->

num)

{fun^ {p : (num

->^ num)} {p 0}}} {with {snd

: ((num

->^ num) -> num) {fun^ {p : (num

->^ num)} {p 1}}} {snd^ {{pair

1} 2}}}}}

Pairs

{with {pair : (bool

->^ (bool

->^ (num

-> bool)))

{fun^ {x : bool}{fun

{y : bool}{fun^

{s : num}{if0^ s x y}}}}}

{with {fst : ((num

->^ bool) ->

bool)

{fun^ {p : (num

->^ bool)} {p 0}}} {with {snd

: ((num

->^ bool) ->

bool)

{fun^ {p : (num

->^ bool)} {p 1}}} {snd {{pair

true} false}}}}}